singular perturbation

The paper deals with the Korteweg-de Vries equation with variable coefficients and a small parameter at the highest derivative.  The asymptotic step-like solution to the equation is obtained by the non-linear WKB technique.  An algorithm of constructing the higher terms of the asymptotic step-like solutions is presented.  The theorem on the accuracy of the higher asymptotic approximations is proven.  The proposed technique is demonstrated by example of the equation with given variable coefficients.  The main term and the first asymptotic approximation of the given example are found, their a

The paper deals with the singularly perturbed Korteweg–de Vries equation with variable coefficients.  The equation describes wave processes in various inhomogeneous media with variable characteristics and small dispersion.  We consider the general algorithm of construction of asymptotic solutions of soliton type to the equation and present its approximate solutions of this type.  We analyze properties of the constructed asymptotic solution depending on a small parameter.  The results are demonstrated by the examples of the studied equation.  We show that for an adequate description of quali